A list of the world’s fastest super computers has been released and Japan tops it with a computer that is three times faster than any other computer on Earth. I wonder if this kind of computer power would help with my terrible internet connection.

The K Computer is based at the RIKEN Advanced Institute for Computational Science in Kobe, Japan, and smashes the previous supercomputing records with a processing power of more than 8 petaflop/s (quadrillion calculations per second) - three times that of its nearest rival.

No, computer processing speed is mostly irrelevant to internet bandwidth. In-fact, it's mostly irrelevant to a PCs over all performance. That the central processing unit (CPU) can perform many quadrillion flops per second is irrelevant if the North Bridge is not capable of buffering at the same speed. If the bands leading from the peripheral components interfaces (PCI slots) or advanced graphics ports (AGP) is slow then the processor will process information fast, but will bottle-neck at the South Bridge. A new MB architecture would be required, the AT form-factor with its typical bus speeds would not allow for much performance increase. Also you would need to make sure your Network Interface Card (NIC) is top of the line and operating at 1GB/s or greater. Gigabit Ethernet NICs are becoming more common, but I personally am running a 100 Base-TX network with only a 10/100 full-duplex NIC.